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Mirrors > Home > ILE Home > Th. List > 2omotap | GIF version |
Description: If there is at most one tight apartness on 2o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.) |
Ref | Expression |
---|---|
2omotap | ⊢ (∃*𝑟 𝑟 TAp 2o → EXMID) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2omotaplemst 7256 | . . . . 5 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝑥 = {∅}) → 𝑥 = {∅}) | |
2 | 1 | ex 115 | . . . 4 ⊢ (∃*𝑟 𝑟 TAp 2o → (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅})) |
3 | df-stab 831 | . . . 4 ⊢ (STAB 𝑥 = {∅} ↔ (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅})) | |
4 | 2, 3 | sylibr 134 | . . 3 ⊢ (∃*𝑟 𝑟 TAp 2o → STAB 𝑥 = {∅}) |
5 | 4 | adantr 276 | . 2 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅}) |
6 | 5 | exmid1stab 4208 | 1 ⊢ (∃*𝑟 𝑟 TAp 2o → EXMID) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 STAB wstab 830 = wceq 1353 ∃*wmo 2027 ⊆ wss 3129 ∅c0 3422 {csn 3592 EXMIDwem 4194 2oc2o 6410 TAp wtap 7247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-tr 4102 df-exmid 4195 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-1o 6416 df-2o 6417 df-pap 7246 df-tap 7248 |
This theorem is referenced by: exmidmotap 7259 |
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